Contents

Abstract

1. Introduction

From the theoretical point of view, the magnetosphere is a direct
result of impenetrability or frozen-in magnetic field condition
of two highly conducting magnetoplasmas. Hereby, the magnetopause
can be interpreted as a tangential discontinuity across which
pressure is in balance. Hence, as investigated first by
Mead and Beard [1964],
the standoff distance of the magnetopause can be
determined from pressure balance. More specifically, the magnetopause
is located at a distance where the planetary magnetic field
pressure (the particles make only a negligible contribution)
equals the dynamic pressure of the solar wind (neglecting the
small contribution of the interplanetary magnetic field (IMF)). It
is clear that when parameters defining the pressure balance
change, the position of the magnetopause will also vary.

It emerges from the above that the primary source of magnetopause
motion is a change in
dynamic pressure of the
solar wind.
However, there
is another source that makes the magnetopause move earthward even
when the dynamic pressure is constant. This phenomenon is called
"erosion" and was identified in the 1970s when
magnetopause crossings made by the OGO 5 spacecraft were investigated
[Aubry et al.,1970;
Fairfield, 1971].
Recent observational
signatures of erosion in the inner magnetosphere have been
reported by
Sibeck [1994] and
Tsyganenko and Sibeck [1994].

Quite generally, one can say that erosion happens during intervals
when the IMF has a persistent southward component,
Bz<0.
Furthermore, the amount of erosion depends on the strength of this
north-south component and can be of the order of 1
RE for
every 5 nT
Bz negative
[Kawano and Russell, 1997].
With this
in mind, models of magnetopause shape have been developed that
account for both the solar wind dynamic pressure and the IMF
Bz component
[Petrinec and Russell, 1996;
Roelof and Sibeck, 1993;
Shue et al.,1997, 1998].
For fixed dynamic pressure, these
empirically based models give the displacement of the
magnetopause resulting from the IMF
Bz.

Since a persistent southward orientation of the IMF is a
prerequisite for magnetic reconnection, a physical
connection between the erosion and reconnection phenomenon was
established. The reconnection process "opens" terrestrial field
lines; that is, as a result, magnetic field lines exist with one
"end" in the ionosphere and another in the flowing magnetosheath
plasma, that is ultimately in the solar wind
[Dungey, 1961].
Indeed,
many features of erosion can be modeled in terms of a
reconnection model. Nevertheless, one can say in general terms
that the physics of dayside magnetosphere erosion is still only
partially understood.

So far, the analysis of erosion leads only to the identification of
several distinct features, which in turn lead to the development
of specific model approaches. These can be summarized as
follows:

Reconnection leads to a transfer of magnetic flux
from the dayside to the nightside magnetosphere, where it builds
up during the so-called growth phase of a substorm prior to
release at substorm onset. On the dayside, magnetic field
intensity decreases near the subsolar point, pressure balance is
violated, and a motion of the magnetopause toward Earth ensues
[Holzer and Slavin, 1978].
Possibly when an instability threshold
is reached,
[Baker et al.,1984,
McPherron et al.,1973]
reconnection starts in the geomagnetic tail, which returns
magnetic flux to the dayside. When the reconnection rates are
balanced, the magnetosphere reaches a new equilibrium position and
shape, and erosion has stopped.

Maltsev and Lyatsky [1975]
and
Sibeck et al. [1991]
proposed a model in which erosion is
interpreted as the effect of a Birkeland current loop in the cusp
region. When the IMF
Bz turns southward, the strength of the
region 1 Birkeland currents increases. The fringe fields of these
Birkeland currents act to reduce the magnetic field strength
within the outer dayside magnetosphere. Thus, when the
interplanetary magnetic field turns southward, the dayside
magnetopause moves inward to restore pressure balance.

Kovner and Feldstein [1973]
attributed erosion to the penetration
of the magnetic field from the magnetosheath to the magnetosphere.
Their hypothesis was subsequently developed further by
Pudovkin et al. [1998],
who assumed that field penetration is associated
with magnetic field reconnection.

Hence, at present, although various aspects of magnetopause
erosion have been studied, a global picture of the detailed
physics behind this phenomena awaits elaboration.

A goal of this paper is to analyze the above-mentioned approaches
and put them into a global perspective. To
correctly
understand
the effects of magnetospheric erosion, it is necessary -- in our
view -- to take into account a variety of interrelated phenomena,
including the appearance of the magnetic barrier near the
magnetopause, unsteady (bursty) reconnection of the IMF and the
magnetospheric magnetic field, and consequences of reconnection
such as the penetration of the IMF to the magnetosphere. Only
after a careful investigation of all these effects is it possible
to analyze pressure balance and investigate erosion
phenomenon in detail.

2. Unsteady Reconnection

Figure 1

In the beginning of this paper
we do not consider the magnetopause in all its
details but
instead
pay attention to a simple theoretical system
consisting of a three-dimensional current sheet separating two
uniform and identical plasmas with oppositely directed magnetic
fields (Figure 1a).

Due to the frozen-in condition, the plasma and magnetic field from
different sources do not intermix and occupy separate regions,
magnetosphere and magnetosheath, for example, and in our simple
case, half spaces above and below the current sheet. From the
physical point of view, it is clear that free energy cannot be
accumulated infinitely by adding magnetic flux from outside
and storing it at the current sheet. Eventually, the gradients
there become so sharp that the plasma loses the frozen-in
property, and this is the starting point for reconnection.

Since
current sheets such as the magnetopause are highly non-uniform
in nature,
it seems reasonable to suppose that the breakdown of
the frozen-in approximation occurs locally in the current sheet
due to some kind of dissipation mechanism rather than
instantaneously throughout the whole length. This natural
assumption distinguishes Petschek-type reconnection (local
dissipation) from magnetic field annihilation theories such as
magnetic field diffusion
[Parker, 1963;
Sweet, 1958]
or the
tearing instability
[Galeev et al.,1986].
For space and
astrophysical applications, pure magnetic field diffusion is too
slow to be of interest, but, as was first realized by
Petschek [1964],
those disturbances can be transmitted through the plasma via
large-amplitude MHD waves or shocks. These waves rapidly escape
from the dissipative region where reconnection is initiated, transfer
the reconnection-associated disturbances to other parts of the
current sheet, and establish an outflow region for the plasma
streaming toward the current sheet (Figure 1b). Plasma entering
the outflow region is accelerated and heated at the slow shocks
and then collected inside. The outflow region is also referred
to as the field-reversal region, since it connects magnetic field
lines across the current sheet, thus establishing a topologically
new region of reconnected flux. The leading front of the outflow
region is propagated along the current sheet with Alfvén
velocity, and therefore the size of the outflow region rapidly
outgrows that of the diffusion region, so that the former
provides the dominant means of converting and transporting energy
and momentum during the reconnection process
[Semenov et al.,1983].

So far, we have described the switch-on phase of reconnection,
when
a dissipative electric field is generated in the diffusion region.
But Petschek's wave mechanism does not operate continuously and at
all times. At some stage, reconnection should switch off (Figure 1c),
in which case no more reconnected flux is added to the
system. The slow shocks and the separatrices, which bound the
reconnected flux tubes, detach from the former site of diffusion
at the time of switch-off. Since the diffusion region no longer
acts as a generator of a dissipative electric field and MHD waves,
the outflow region will also detach from the reconnection site,
and it propagates like a pair of solitary waves in opposite
directions along the current sheet. But the outflow region
cannot be considered as a soliton, because the slow shocks previously
generated continue to propagate toward the edges of the
current sheet and to accelerate and heat plasma so that
all plasma inside the reconnected tubes bounded by separatrices
will be trapped inside the outflow region. Therefore, the outflow
region continues to change shape and increase in size even though
no more reconnected flux is added.

During the switch-on phase in the immediate vicinity of the
diffusion region, the structure of magnetic field and plasma flow
is very similar to the original Petschek model (Figure 1b) and,
indeed, it can be shown that the time-dependent reconnection
solution tends to Petschek's solution in this limit
[Pudovkin and Semenov, 1985;
Semenov et al.,1983].
So, we can say that our
model is an extension of Petschek's wave mechanism for the case of
a time-varying reconnection rate. On the other hand, the global
structure differs quite a lot from Petschek's picture
depicted in Figure 1d,
in particular, during the switch-off phase. We believe
that a time-dependent reconnection model is more useful for applications
than the original Petschek model of steady-state reconnection,
since nearly all manifestations of reconnection
are strongly time-dependent
in nature
and even are explosive in character.

We do not know enough about the way in which reconnection is initiated,
but we can describe the large-scale consequences of the locally
initiated reconnection process by adopting a semi-phenomenological
approach. In this approach, we model the initiation
introducting a reconnection electric field
E*(r,t) inside
the diffusion region rather than by specifying a concrete dissipative
process. In addition, we assume, like Petschek, that the diffusion
region is very small compared to the size of the system, so that
in rough approximation, this region corresponds to the so-called
reconnection or X-line. By the way, this is an example of a
commonly used technique whereby non-ideal effects are lumped into
discontinuities to work in the framework of ideal MHD.
So, we are describing the diffusion region
and the behavior of the dissipative process inside in terms of an
initial-boundary condition to solve ideal MHD equations, and
this initial-boundary condition corresponds to the specification
of the X-line and the behavior of the reconnection rate
E*(r,t) along it
[Rijnbeek and Semenov, 1993].

The solution of the reconnection problem in an incompressible plasma
in the dimensionless form can be presented as follows
[Biernat et al.,1987;
Pudovkin and Semenov, 1985;
Semenov et al.,1983].

(1)

(2)

(3)

where (1) is the plasma velocity; (2) is the magnetic
field inside the
field reversal region (FRR); (3) is the shape of Petschek shock
for the first quadrant;
E*(t,y) is the electric field along
the reconnection line, or so-called reconnection rate. All quantities
are normalized to the initial magnetic field
B0, the initial
Alfvén velocity
vA=B0/(4pr)1/2,
the length of the
reconnection line
L, the time
L/vA, the pressure
B0/4p,
and the Alfvén electric field
EA=B0vA/c.
Here
e =E*/EA1 is a small parameter.

Figure 2

The first order corrections to the
x component of the magnetic field
and to the total (gas + magnetic) pressure in the inflow region
(IR) can be obtained from the Poisson integrals

(4)

where

(5)

(6)

The whole solution (1)-(6) is defined by the
reconnection rate
E*(t,y). To model bursty reconnection, we
can use series of pulses; one is shown in Figure 2.

3. Current Sheet Motion

Figure 3

Reconnection leads to transfer of magnetic flux from the
reconnection site along the current sheet (see Figure 1). As a
consequence, magnetic field intensity weakens near the
diffusion region, and hence the total pressure decreases
also. The behavior of disturbances
Bx(1)(t) and
P(1)(t) near the diffusion region (at
x=y=0, z=0.3 ) is shown in
Figure 3.
It can be seen that the reconnection event
also
produces
the negative pressure pulse in the vicinity of the reconnection
site. The electric field in the diffusion region
is
first
switched on, reaches its maximum value, and then decreases
(Figure 2).
Similarly, total pressure begins from the background
value, reaches its minimum value, and increases when the FRRs run
away during the switch-off phase (Figure 3). Asymptotically, the
behavior of the total pressure disturbance during this last stage
is as follows

where
F0 is the reconnected magnetic flux.

Such pressure variations happen from both the magnetosheath
(sh) and the magnetosphere (mg) sides of the current sheet for the
symmetric model used so far

(7)

and therefore, they can not lead to motion of the magnetopause. For
motion to start, evidently some kind of asymmetry has to
appear, and, indeed, physical conditions in the magnetosheath and
in the magnetosphere are highly different. First of all, in the
magnetosheath there is powerful solar wind flow. Second, there is
bow shock at which dynamic pressure remains unchanged at
P sh(0)= const
for the erosion events under consideration.
Therefore, disturbances produced by reconnection at the
magnetopause have to propagate against solar wind flow and then
reflect from the bow shock. In the magnetosphere, there is
neither bow shock nor strong plasma flow; hence, evolution of
pressure disturbances produced by reconnection must be different
in the magnetosheath and inside the magnetosphere.

Bearing these circumstances in mind, we can believe that total
pressure from the magnetosheath side of the diffusion region has
to tend to asymptotic value
P sh(0)= const
more quickly than
from the magnetosphere side.

Theoretically,
it is rather difficult to
determine exactly how fast
P shP sh(0),
but
we can make a simple estimation
for the first consideration.
We suppose that
P shP sh(0)
with some characteristic time
t sh, so that
pressure balance at the magnetopause takes the form

(7a)

A pressure pulse decays mostly due to the fast mode wave; therefore,
tsh can be estimated approximately as time propagation
of the
fast wave
vs from subsolar point to the bow shock
t sh=L sh/vs, where
L sh 3RE
is the width of the
magnetosheath. The relaxation time
t sh also
can be
estimated
from the exact solution of unsteady annihilation
[Heyn and Pudovkin, 1993]
or from numerical simulation
[Pudovkin and Samsonov, 1994].

The main idea of our model is that the breach in total pressure
made by reconnection is closed faster from the magnetosheath side due
to the powerful flow of the solar wind than from the magnetosphere side.
This leads to an asymmetry in pressure balance and earthward motion of the
magnetopause after each pulse of reconnection. The
position of the magnetopause can be easily found from (7a)

(8)

where the variation of total pressure has to be taken from the
equation (4), and initial position is supposed to be
r=10 RE. We can include
N pulses of reconnection with time
repetition
tr

(9)

We take the following initial parameters for the numerical
calculation: The half-length of the reconnection line is
L = 3 RE, the Alfvén
velocity is
vA=300 km s
-1 as average of Alfvén
velocity in the magnetosphere and in the magnetosheath,
the reconnection rate is
e = 0.3, the pressure relaxation time
is
t sh = 3 min, the repetition time of reconnection pulses is
tr = 5 min.

Figure 4

The resulting stand-off distance of the magnetopause as a
function
of time is shown in Figure 4 (upper dotted line). It can be seen
that each reconnection event leads to a jump-like motion to the
Earth with a more smooth return nearly to the same level afterwards.
It turns out that on average, the magnetopause shifts less than 1/4
RE per hour. The contribution of all reconnection pulses
is not
enough for the observed 1
RE erosion of the magnetopause,
because each pressure pulse decreases rather fast
1/t2 (see (7)).

We still did not take into account the appearance of region 1
field-aligned currents. It is well known that a reconnection event
generates field aligned currents
[Pudovkin and Semenov, 1985].
As
far as an FR-region moves along the current sheet (the magnetopause
in our case), the contribution of this current system is included
automatically in the pressure behavior near the diffusion region,
but when reconnection-associated disturbances turn off from the
current sheet (turn off from the magnetopause to the ionosphere in
the cusp region), we have to take into account the contribution of
the field-aligned current system separately.

Generally speaking, this problem is rather difficult, because we
have to find a field-aligned current from each reconnection pulse,
propagation of the field-aligned current in the form of an Alfvén
wave from the cusp to the ionosphere, and then determine
the contribution of this current system to pressure balance near
the subsolar point. For the simple model under consideration, we
will not attempt to solve this difficult problem but instead
try to estimate the contribution of the region 1 field-aligned current
system.

In our model, the FR-region propagates toward the cusp region and then
turns off to the ionosphere along a magnetic field line. The
pressure variation
P mg(1)(t) can be calculated from
equation (4) until the FR-region reaches the cusp. Our
suggestion is that the contribution of the field-aligned current
to the pressure balance at the subsolar point is constantly of the
order of
P mg(1)(t cusp), where
t cusp is the time
propagation of the FR-region from the diffusion region to the
cusp. This implies that we suppose that for each reconnection
event for
t cusp the pressure disturbance is determined by
(4) and then keeps const
=P mg(1)(t cusp) for
t>t cusp.

The result of magnetopause erosion based on this assumption
is
shown in Figure 4
(solid line). The jump-like behavior of the
magnetopause motion is the same as previously, but the average
shift is much bigger,
1 RE. Therefore, the
details (jumps)
of magnetopause motion are determined by bursty reconnection, but
erosion itself mostly depends on the strength of the region 1
field-aligned current system.

4. Discussion

The model described above naturally incorporates all three
approaches to magnetopause erosion.

(1) Flux transfer
[Holzer and Slavin, 1978].
Our model is
based on the solution of the impulsive reconnection problem, which
is determined by the reconnection electric field
E*(t,y) (see (1)-(6)).
The main physical reasons for the pressure
pulse and the region 1 field-aligned current system are reconnection
events and the transfer of reconnected flux from the dayside to the
nightside of the magnetosphere.

(2) Penetration of the magnetic field from the magnetosheath
to the magnetosphere
[Kovner and Feldstein, 1973;
Pudovkin et al., 1998].
Let us rewrite pressure balance in terms of magnetic
fields. For our simple model with a symmetric current sheet, we can
suppose that the gas pressure in the magnetosheath and in the
magnetosphere is the same; hence, the pressure balance is the
following:

(10)

After each reconnection event, the second term on the right-hand
side of equation (10) is small for
t>t sh. This implies
that
magnetosheath magnetic field
effectively
penetrates
the magnetosphere at a relaxation time scale
t>t sh but
magnetospheric field does not penetrate into the magnetosheath. It
cannot be emphasized enough that no magnetic charges appear, and
the same reconnected flux is subtracted from the magnetosphere and
the magnetosheath initial flux. Penetration of magnetic field
from the magnetosheath to the magnetosphere needs to be understood
effectively as a consequence of different pressure pulse evolution
in the magnetosheath and magnetosphere as it was described
above. The bow shock and pile-up process (magnetic barrier) make
an asymmetry in pressure pulse propagation, which leads to a jump-like
motion of the magnetopause. Therefore, this effect is the essential
component of the magnetopause erosion theory.

(3) Field-aligned currents
[Maltsev and Lyatsky, 1975;
Sibeck et al., 1991].
Erosion is most often interpreted as
an effect of the region 1 Birkeland current system. It turns
out that the fringe fields of these Birkeland currents
reduce magnetic field strength near the subsolar point, and as
a consequence, the dayside magnetopause moves inward to rebuild
pressure balance.

Compared with the pressure pulse effect (or the effect of
penetration of the magnetic field from the magnetosheath to the
magnetosphere), the contribution of field-aligned currents from a
reconnection event is rather small. But it is important to note
that the effects of field-aligned currents from several
reconnection pulses are accumulated. Therefore, after each
reconnection event the magnetopause first quickly moves inward and
then reverts to a position slightly shifted to the Earth
(see Figure 4).
The difference between forward and backward motion of the
magnetopause is the effect of the region 1 Birkeland current system.
Hence, time averaged erosion mostly depends on
field-aligned currents.

5. Conclusions

1. A simple model of magnetopause erosion based on analytical
impulsive reconnection theory is presented.

2. It is shown that bursty reconnection leads to an inward jump-like
motion of the magnetopause.

3. The model incorporates all three most popular approaches:
Flux transfer, penetration of the magnetic field from
magnetosheath to magnetosphere, and the effect of a
region 1
Birkeland current system.
The first two effects are responsible
for the jump-like motion of the magnetopause, and the last one
is responsible for
the shift of the magnetopause to Earth.

Acknowledgments

Part of the work was done while V.S.S. and N.V.E. were at the Space
Research Institute of the Austrian Academy of Sciences in Graz, and while H.K.B.
and S.M. were at the Institute of Physics of the State University of St. Petersburg.
This work is partially supported by INTAS-ESA project 99-01277, by the INTERGEOPHYSICS
programme of the Russian Ministry of Higher Education, by the Austrian "Fonds zur
Förderung der wissenschaftlichen Forschung''
under project P13804-TPH, by grant No. 98-05-65290 of the Russian Foundation
of Basic Research, by grant No. 97-0-13.0-71 of the Russian Ministry of Education,
and by the Austrian Academy of Sciences "Verwaltungstelle für Auslandsbeziehungen."